You can catch bugs with a Bug Net. To catch bugs, you often need to sneak up on them. To do so, arm your Bug Net and hold A a good distance from the bug. This allows you to sneak up on the bug. Release A to spring your net and you'll have a new insect buddy. Note that some bugs will be disturbed and fly away (especially beetles) if you walk quickly near them. For these bugs you must push the analog slider very slightly towards the bug to creep up on it with your Bug Net raised.

Once caught, they will automatically be registered in the Encyclopedia. After you have a bug in your inventory, you can do the following:

This detailed list shows the value, rarity, location, and availability of all 72 bug species. It follows the same order as the in-game menu. Rarity is given in numbers 1-7, with 1 being rarest and 7 being extremely common.